\documentclass[style=kuleuven]{powerdot}
\usepackage[dutch]{babel}
\usepackage{tikz, bibentry, amssymb, color}
\usetikzlibrary{shapes}

\title{Self-Avoiding Walks met alternerende staplengtes}
\author{\color{pdcolor3}Koen Van den dries \& Jonas Vanthornhout\\\color{pdcolor3}Begeleider: prof. dr. B. Demoen}
\date{\color{pdcolor3}4 mei 2011}

\pdsetup{theslide=\arabic{slide}}

\begin{document}
\maketitle

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\section{Self-Avoiding Walk}
\begin{slide}{Wat is een SAW?}
	\vspace*{\fill}
	\begin{center}
		\begin{tabular}{ccc}
			Random Walk & \hfill &\onslide*{2-}{Self-Avoiding walk} \\ \hfill \\
			\begin{tikzpicture}[node distance = 2cm]
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			\end{tikzpicture}
			& \hfill &\onslide*{2-}{
			\begin{tikzpicture}[node distance = 2cm]
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			\end{tikzpicture}} \\
		\end{tabular}
	\end{center}
	\vspace*{\fill}
\end{slide}

\begin{slide}{Definitie}
	\vspace*{\fill}
	\begin{tikzpicture}
		\fill[rounded corners, fill=KULBlue] (0,4.1) rectangle (9.5,5);
		\color{white}
		\pgftext[base,left, x=0.5cm, y=4.45cm]{Self-avoiding walk (SAW)}
		\fill[rounded corners, fill = KULblue] (0,2) rectangle (9.5,4);
		\color{black}
		\pgftext[base, left, x=0.5cm, y=3.5cm]{Een random walk die op geen enkel punt zijn afgelegde pad}
		\pgftext[base, left, x=0.5cm, y=3cm]{snijdt of raakt. In tegenstelling tot de random walk heeft een}
		\pgftext[base, left, x=0.5cm, y=2.5cm]{SAW een eindige lengte.}
	\end{tikzpicture}
	\vspace*{\fill}
\end{slide}


\begin{slide}{Eindige lengte}
	\vspace*{\fill}
	\begin{center}
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		\end{tikzpicture}
	\end{center}
	\vspace*{\fill}
\end{slide}

\subsection{m-n SAW}
\begin{slide}{m-n SAW}
	\vspace*{\fill}
	\begin{center}
		Staplengtes: $m\,\&\,n$ \\ \hfill \\
		\begin{tikzpicture}[node distance = 1cm]
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		\end{tikzpicture} \\ \hfill  \\
		Alternerend \& random
	\end{center}

	\vspace*{\fill}
\end{slide}

\section{Verwant onderzoek}
\begin{slide}{Verwant onderzoek}
	\vspace*{\fill}
	\begin{itemize}
		\item vooral onderzoek naar aantal SAW's
		\item afstand tussen eindpunten
		\item gemiddelde lengte is 70.7 $\pm$ 0.2 \cite{hemmer, diplomarbeit}
	\end{itemize}
	\includegraphics[scale = 0.9, trim = 10mm 150mm 110mm 58mm, clip]{fit_diplomarbeit}
	\vspace*{\fill}
\end{slide}

\section{1-1 SAW}
\begin{slide}{Wat?}
	\vspace*{\fill}
	\begin{itemize}
		\item alternerende $1-x$ SAW
		\item eerste staplengte $=1$
		\item tweede staplengte $\approx 1$
	\end{itemize}
	Vraag: be\"invloedt dit het gemiddeld aantal stappen?
	\vspace*{\fill}
\end{slide}

\begin{slide}{Resultaten}
	\vspace*{\fill}
	\onslide*{1}{
	\begin{itemize}
		\item $1-x$ met $x \lessapprox 1:\,\overline{lengte}=53.37 \pm 0.04$
		\item $1-x$ met $x \gtrapprox 1:\,\overline{lengte}=52.83 \pm 0.04$
		\item $1-1:\,\overline{lengte}=70.77 \pm 0.02$
	\end{itemize}}
	\onslide*{2}{
		\includegraphics[scale=0.75]{1-1}
	}
	\vspace*{\fill}
\end{slide}

\begin{slide}{Verklaring}
	\vspace*{\fill}
	\onslide*{1}{
		2 ongeveer gelijke staplengtes \\ \vspace*{1em}
		$\Rightarrow$ SAW kan dichter bij pad lopen
	}
	\onslide*{2}{
		\begin{center}
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			\end{tikzpicture}
		\end{center}
	}
	\vspace*{\fill}
\end{slide}

\begin{slide}{Conclusie}
	\vspace*{\fill}
	\begin{itemize}
		\item slecht geconditioneerd
		\item gemiddelde lengte $\downarrow$ met minimale aanpassing
	\end{itemize}
	\vspace*{\fill}
\end{slide}

\section{1-$\infty$ SAW}
\begin{slide}{Wat?}
	\vspace*{\fill}
	\begin{itemize}
		\item alternerende $1-x$ SAW
		\item \'e\'en staplengte $=1$
		\item andere staplengte $\gg1$
	\end{itemize}
	Vraag: welke invloedt heeft dit op het gemiddeld aantal stappen?
	\vspace*{\fill}
\end{slide}

\begin{slide}{Resultaten}
	\vspace*{\fill}
	\begin{center}
		\onslide*{1}{
			\includegraphics[scale=0.75]{1-infty-lin}
			fit: $l(x) = ax+b$ \\
			$a=0.225032$ \\
			$b=78.2219$
		}
		\onslide*{2}{
			\includegraphics[scale=0.75]{1-infty-bult}
		}
	\end{center}
	\vspace*{\fill}
\end{slide}

\begin{slide}{Verklaring}
	\vspace*{\fill}
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	\onslide*{1}{
		Stijgende lengte komt door een speciaal geval \\
		\hspace{2em} SAW maakt rechthoek \\
		\hspace{2em} SAW gaat deze rechthoek binnen
	}
	\begin{center}
		\begin{tikzpicture}[node distance = 1.5cm, auto]
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			}
		\end{tikzpicture} \\
		\onslide*{6-}{
			$ hoogte \approx x \quad breedte = x $ \\
			$\Rightarrow \#knopen = 2x$
		}
	\end{center}
	\vspace*{\fill}
\end{slide}

\begin{slide}{Conclusie}
	\vspace*{\fill}
	\begin{itemize}
		\item gemiddelde lengte $\uparrow$
		\item stijging vooral door speciaal geval
		\item inherent aan alternerende SAW
	\end{itemize}
	\vspace*{\fill}
\end{slide}

\section{Even-Oneven}
\begin{slide}{Wat}
	\vspace*{\fill}
	\begin{center}
		\onslide*{1}{
			\includegraphics[scale=0.75]{1-infty-bult}
		}
		\onslide*{2}{
			\includegraphics[scale=0.75]{1-infty-bult}
			\begin{tikzpicture}[remember picture, overlay]
				\draw (-2.75,3.4) node[minimum height=5cm, minimum width=1.5cm, draw, very thick] {};
				\draw (2.2,1.5) node[minimum height=1.3cm, minimum width=1cm, draw, very thick] {};
		\end{tikzpicture}}
	\end{center}
	\vspace*{\fill}
\end{slide}


\begin{slide}{Detail}
	\vspace*{\fill}
	\begin{center}
		\onslide*{1}{
			kleine oneven lengtes
			\includegraphics[scale=0.7]{bultfitKO}
		}
		\onslide*{2}{
			kleine even lengtes
			\includegraphics[scale=0.7]{bultfitKE}
		}
		\onslide*{3}{
			grote even lengtes
			\includegraphics[scale=0.7]{bultfitGE}
		}
		\onslide*{4}{
			grote oneven lengtes
			\includegraphics[scale=0.7]{bultfitGO}
		}
	\end{center}
	\vspace*{\fill}
\end{slide}

\begin{slide}{Wat}
	\vspace*{\fill}
	\begin{center}
		Wat is de oorzaak van dit grote verschil?
	\end{center}
	\vspace*{\fill}
\end{slide}

\begin{slide}{Verklaring}
	\vspace*{\fill}
	\onslide*{1-}{
		eerste bult: oneven grootst \\
		$\Rightarrow$ laatste stap is $1$ \\ \hfill \\
	}
	\onslide*{2}{
		tweede bult: even grootst \\
		$\Rightarrow$ laatste stap is $\gg1$ \\
	}
	\vspace*{\fill}
\end{slide}

\section{Toepassingen}
\begin{slide}{Polymeren}
	\vspace*{\fill}
	SAW's kunnen gebruikt worden voor polymeren \cite{polymer} \\
	Copolymeren bestaan uit verschillende monomeren \\
		$\Rightarrow$ monomeren kunnen ook random
	\vspace*{\fill}
\end{slide}

\begin{slide}{Copolymeren}
	\vspace*{\fill}
	2 monomeren: \\
	\begin{tikzpicture}
		\draw (0,0) node[circle, fill, black, draw] {};
		\draw (1,0) node[diamond, fill, red, draw] {};
	\end{tikzpicture}
	\\Alternerend: \\
	\begin{tikzpicture}
		\draw (0,0) .. controls (1,1) and (3,1) .. (4,0)
		\foreach \p in {0,0.25,...,1} {node[pos=\p, circle, draw, fill, black, scale=0.5]{}}
		\foreach \p in {0.125,0.375,...,1}  {node[pos=\p, diamond, draw, fill, red, scale=0.5]{}};
		\draw (4,0) .. controls (5,-1) and (6,-1) .. (7,0)
		\foreach \p in {0,0.25,...,1} {node[pos=\p,circle, draw, fill, black, scale=0.5]{}}
		\foreach \p in {0.125,0.375,...,1}  {node[pos=\p, diamond, draw, fill, red, scale=0.5]{}};
	\end{tikzpicture}
	\\Random: \\
	\begin{tikzpicture}
		\draw (0,0) .. controls (1,1) and (3,1) .. (4,0)
		\foreach \p in {0,0.375,0.5,0.625,1} {node[pos=\p, circle, draw, fill, black, scale=0.5]{}}
		\foreach \p in {0.125,0.25,0.75,0.875}  {node[pos=\p, diamond, draw, fill, red, scale=0.5]{}};
		\draw (4,0) .. controls (5,-1) and (6,-1) .. (7,0)
		\foreach \p in {0.25,0.375,0.5,0.875} {node[pos=\p, circle, draw, fill, black, scale=0.5]{}}
		\foreach \p in {0.125,0.625,0.75,1}  {node[pos=\p, diamond, draw, fill, red, scale=0.5]{}};
	\end{tikzpicture}
	\vspace*{\fill}
\end{slide}

\begin{slide}{Toepassingen}
	\vspace*{\fill}
	\begin{itemize}
		\item SAW gebruiken als simulatie
		\item SAW als voorspeller polymeren
		\item SAW voor reverse-engineering polymeren
	\end{itemize}
	\vspace*{\fill}
\end{slide}

\section{Andere resultaten}
\begin{slide}{Andere resultaten}
	\vspace*{\fill}
	\tableofcontents[content=currentsection]
	\vspace*{\fill}
\end{slide}

\subsection{Hogere dimensies}
\begin{slide}{Hogere dimensies}
	\vspace*{\fill}
	\begin{center}
		$1-1$ SAW \\
		\begin{tabular}{|c|c|c|}
			\hline
			Dimensie & Stappen \\ \hline
			2D & $70.77 \pm 0.02$ \\ \hline
			3D & $3971.07 \pm 6.8$ \\ \hline
			4D & $>3654512.40 \pm 212653.95$ \\ \hline
		\end{tabular}
	\end{center}
	\vspace*{\fill}
\end{slide}

\subsection{2 kanten groeien}
\begin{slide}{2 kanten groeien}
	\vspace*{\fill}
	\begin{center}
		\begin{tikzpicture}[node distance = 1.5cm]
			\node[node] (node1) {};
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			\node[nodered, left of=node14] (node15) {};
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			\node[nodered, below of=node16] (node17) {};
			\node[nodered, right of=node17] (node18) {};
			\node[nodered, right of=node18] (node19) {};

			\draw (node1) -- (node2) -- (node3) -- (node4) -- (node5) -- (node6) -- (node7) -- (node8) -- (node9) -- (node10) -- (node11);
			\draw[red] (node11) -- (node12) -- (node13) -- (node14) -- (node15) -- (node16) -- (node17) -- (node18) -- (node19);
		\end{tikzpicture} \\
		$\overline{lengte\,symm} = 120.67 \pm 0.07$ \\
		$\overline{lengte\,asymm} = 119.21 \pm 0.07$
	\end{center}
	\vspace*{\fill}
\end{slide}

\subsection{Random}
\begin{slide}{Random}
	\vspace*{\fill}
	\onslide*{1}{
		\begin{center}
			\includegraphics[scale=0.75]{rand}
			$\overline{lengte} = 58.20 \pm 0.04$
		\end{center}
	}
	\onslide*{2}{
		Verklaring: \\
		\indent rechthoek kan bijna niet gevormd worden \\
		$\Rightarrow$ geen tweede bult
	}
	\vspace*{\fill}
\end{slide}

\subsection{Andere geometrie\"en}
\begin{slide}{Andere geometrie\"en}
	\onslide*{1}{
		\vspace*{\fill}
		\begin{center}
			triangulair \\
			\begin{tikzpicture}[node distance = 1.5cm]
				\node[node] (node1) {};
				\node[node, below right of=node1] (node2) {};
				\node[node, below left of=node2] (node3) {};
				\node[node, below right of=node3] (node4) {};
				\node[node, draw=none,fill=none, above left of=node4] (node5) {};
				\node[node, below left of=node5] (node6) {};
				\node[node, below right of=node6] (node7) {};
				\node[node, below left of=node7] (node8) {};
				\node[node, above left of=node8] (node9) {};
				\node[node, above left of=node9] (node10) {};
				\node[node, above right of=node10] (node11) {};

				\draw (node1) -- (node2) -- (node3) -- (node4) -- (node6) -- (node7) -- (node8) -- (node9) -- (node10) -- (node11);
			\end{tikzpicture} \\
			$\overline{lengte} = 77.17$
		\end{center}
		\vspace*{\fill}
	}

	\onslide*{2}{
		\vspace*{\fill}
		\begin{center}
			hexagonaal \\
			\begin{tikzpicture}[node distance = 1.5cm]
				\node[node] (node3) {};
				\node[node, right of=node1] (node2) {};
				\node[node, above right of=node2] (node3) {};
				\node[node, right of=node3] (node4) {};
				\node[node, below right of=node4] (node5) {};
				\node[node, below left of=node5] (node6) {};
				\node[node, left of=node6] (node7) {};
				\node[node, below left of=node7] (node8) {};
				\node[node, left of=node8] (node9) {};
				\node[node, above left of=node9] (node10) {};
% 				\node[node, right of=node10] (node11) {};

				\draw (node3) -- (node4) -- (node5) -- (node6) -- (node7) -- (node8) -- (node9) -- (node10) -- (node1)  -- (node2);
			\end{tikzpicture} \\
			$\overline{lengte} = 71.14$
		\end{center}
		\vspace*{\fill}
	}
\end{slide}

\subsection{Obstakels}
\begin{slide}{Obstakels}
	\vspace*{\fill}
	\begin{center}
		\begin{tikzpicture}[node distance = 1.5cm]
			\fill[fill=red] (0,0) rectangle (3,3);
			\node[node] (node1) {};
			\node[node, below of=node1] (node2) {};
			\node[node, right of=node2] (node3) {};
			\node[node, right of=node3] (node3a) {};
			\node[node, below of=node3a] (node3b) {};
			\node[node, right of= node3b] (node3c) {};
			\node[node, above of=node3c] (node3d) {};
			\node[node, right of=node3d] (node3e) {};
			\node[node, below of=node3e] (node4) {};
			\node[node, below of=node4] (node6) {};
			\node[node, right of=node6] (node7) {};
			\node[node, below of=node7] (node8) {};
			\node[node, right of=node8] (node9) {};
			\node[node, below of=node9] (node10) {};
			\node[node, left of=node10] (node11) {};
			\node[node, left of=node11] (node12) {};
			\node[node, above of=node12] (node13) {};
			\node[node, left of=node13] (node14) {};
			\node[node, above of=node14] (node15) {};
			\node[node, left of=node15] (node16) {};

			\draw (node1) -- (node2) -- (node3) -- (node3a) -- (node3b) -- (node3c) -- (node3d) -- (node3e) -- (node4) -- (node6) -- (node7) -- (node8) -- (node9) -- (node10) -- (node11) -- (node12) -- (node13) -- (node14) -- (node15) -- (node16);
		\end{tikzpicture}
	\end{center}
	\vspace*{\fill}
\end{slide}

\subsection{Andere topologie}
\begin{slide}{Andere topologie}
	\vspace*{\fill}
	\begin{center}
		\begin{tikzpicture}
			\draw (-3.5,0) .. controls (-3.5,2) and (-1.5,2.5) .. (0,2.5);
			\draw[xscale=-1] (-3.5,0) .. controls (-3.5,2) and (-1.5,2.5) .. (0,2.5);
			\draw[rotate=180] (-3.5,0) .. controls (-3.5,2) and (-1.5,2.5) .. (0,2.5);
			\draw[yscale=-1] (-3.5,0) .. controls (-3.5,2) and (-1.5,2.5) .. (0,2.5);

			\draw (-2,.2) .. controls (-1.5,-0.3) and (-1,-0.5) .. (0,-.5) .. controls (1,-0.5) and (1.5,-0.3) .. (2,0.2);

			\draw (-1.75,0) .. controls (-1.5,0.3) and (-1,0.5) .. (0,.5) .. controls (1,0.5) and (1.5,0.3) .. (1.75,0);
		\end{tikzpicture} \\
	\end{center}
	\vspace*{\fill}
\end{slide}

\section{Conclusie}
\begin{slide}{Conclusie}
	\vspace*{\fill}
	$1\!-\!x$ SAW\\
	\hspace{4pt} $x \approx 1$: \#stappen $\downarrow$ \\
	\hspace{4pt} $x \gg 1$: \#stappen $\uparrow$\\ \vspace{4pt}
	even $\leftrightarrow$ oneven $\Rightarrow$ verschillende bulten
	\vspace*{\fill}
\end{slide}

\section[tocsection=hidden, slide=false]{Bibliografie}
\nocite{*}
\begin{slide}{Bibliografie}
	\bibliographystyle{alpha}
	\bibliography{bibliography}
\end{slide}

\end{document}